mathematics//decision theory
Decision theory is the branch of mathematics that chooses actions under uncertainty by weighing each action's cost in every possible state of the world by how likely that state is, and it is the step that turns an estimator's output into something a machine does: reject a part, stop a pump, send a drone home. An estimator says how the world probably is; decision theory says what to do about it, usually before anyone is sure and with the clock running.
Decision theory is the branch of mathematics that chooses actions under uncertainty by weighing each action's cost in every possible state of the world by how likely that state is, and it is the step that turns an estimator's output into something a machine does: reject a part, stop a pump, send a drone home. An estimator says how the world probably is; decision theory says what to do about it, usually before anyone is sure and with the clock running.
With actions aaa, states sss known only through a belief P(s)P(s)P(s) (the output of a filter or a classifier) and a cost function C(a,s)C(a,s)C(a,s) for doing aaa when the world is sss, the rule is
a∗=argmina∑sP(s) C(a,s),a^{*} = \arg\min_{a} \sum_{s} P(s)\, C(a,s),a∗=argamins∑P(s)C(a,s),
the action of least expected cost. Its virtue is the clean split between what is believed (the probability) and what matters (the cost); most bad automatic decisions mix the two. A camera judges a part defective with probability ppp. Rejecting a good part costs 3 euros, shipping a bad one 300, so rejecting costs 3(1−p)3(1-p)3(1−p) on average and accepting 300p300p300p: reject whenever p>3/303≈1%p>3/303\approx1%p>3/303≈1%.
The threshold comes from the costs, never from 0.5.
Separate the belief from the cost table and the right cut follows from the ratio of the two error costs, and it moves when the costs move, with nothing retrained (decision threshold).
Before deciding, it may pay to look first, and value of information says how much: an observation is worth what it can change in the decision. When resources can be spent in rounds, acting, observing and only then spending more is sequential allocation.
When each decision changes the state the next one starts from, the frame becomes the Markov decision process and its Bellman equation; when the state cannot be seen, decisions are made on beliefs, and the usual shortcut is certainty equivalence.
Expected value treats many small losses like one total loss. Catastrophic outcomes (a drone falling on people) are therefore imposed as a chance constraint, a bound such as a probability of impact below 10−610^{-6}10−6 per flight hour, and cost is minimized inside it (tails over means). When the other side also decides, against you, the neutral belief gives way to the worst case and the frame is game theory.
The error of a decision lives in five places: the costs, which almost nobody writes down and which dominate the result; the belief (a miscalibrated network moves every threshold, probability calibration); the transition model, which an optimizer exploits wherever it is wrong; the gap between a heuristic and the optimum; and latency. The most expensive error comes before all of them: a question posed badly, a prediction that changes no decision.